The connection
of Lyapunov spectra to fractal dimension is described by Farmer, Ott, and
Yorke [Physica D 7 (1983)]. In two classic papers Bennettin [see my
papers with Posch in Physical Review A 38, 473 (1988) and 39, 2175 (1989)
for complete references] showed how to obtain Lyapunov spectra for
many-body systems. This work generalizes the numerical work of
Spotswood Stoddard and
Joseph Ford
[Physical Review A 8, 1504 (1973)].
Harald Posch and I discovered an elegant Lagrange-Multiplier approach to
this problem [Physics Letters A 113, 82 (1985) and 123, 227 (1987)]
which has later been rediscovered by several others. Our more recent
work on the
Lyapunov Instability of Classical Many-Body Systems
was
reviewed in the fall of 2005 by Harald at an Astrophysics Conference at
Waseda University.
In 2006, as part of a project sponsored by the Academy of Applied Science
(Concord, New Hampshire) at Great Basin College (Elko, Nevada), Carol and I
compared the local values of the forward-in-time and reversed-in-time Lyapunov
exponents with the local extremal phase-space growth rates as determined by a
simple Monte Carlo method. The Lyapunov exponents fluctuate more rapidly than
do the extremal rates. In correspondence with Florian Grond, stimulated by his
2003 and 2005 work in Chaos, Solitons, and Fractals, we developed a better
(cheaper and more nearly accurate) approach based on singular value
decomposition of the local dynamical matrix D [ D = df/dx where dx/dt = f(x) ].
The (real) extremal values turn out to depend only on the symmetrized part of
the matrix. This approach avoids the typical complex eigenvalues described in
our
Polish paper
. The work using singular
values appears in our preprint
(Fall 2006).
The quantum
analog of Lyapunov instability is hard to find beyond the simple
correspondence of wave-packet dynamics with the classical trajectory
(Ehrenfest's Theorem). For an attempt, using the Galton Board model see
[Hoover, Moran, my wife Carol, and Will Evans, Physics Letters A 131,
211 (1988)].